Characteristic not implies isomorph-free in finite group
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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., isomorph-free subgroup)
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Contents
Statement
Statement with symbols
There exists a finite group and a characteristic subgroup
of
such that
is not an isomorph-free subgroup of
. In other words, there exists another subgroup
of
that is isomorphic to
.
Related facts
- Characteristic not implies isomorph-containing
- Characteristic not implies sub-isomorph-free in finite
- Characteristic not implies isomorph-normal in finite
- Characteristic not implies sub-(isomorph-normal characteristic) in finite
- Characteristic not implies injective endomorphism-invariant
Proof
Example of the dihedral group
Further information: dihedral group:D8, subgroup structure of dihedral group:D8, center of dihedral group:D8
Let be the dihedral group of order eight, given as follows, where
denotes the identity element of
:
.
Let be the center of
.
is a subgroup of order two generated by
.
-
is characteristic.
-
is not isomorph-free: The subgroup
of
is isomorphic to
.